Can we use just a script for a driver?
If the property doesn't allow for drivers, it cannot be driven with a driver.

It can be driven through app.handlers though:

import bpy

def my_handler(scene):
    # this code will be run on every frame change
    loc = bpy.data.objects["rigMeenuTeen"].pose.bones["eyes"].location
    bpy.data.textures["mouth shapes"].image_user.frame_offset = int(loc.x)

bpy.app.handlers.frame_change_pre.append(my_handler)